多参数最小支撑树问题的复杂性和算法

建立了多参数最小支撑树问题（ＲＭＳＴ）的模型,并证明该问题是ＮＰ－完全的。利用经典Ｇｒｅｅｄｙ算法,给出了该问题的一个近似算法,并分析了该近似算法的性能比,证明了所给出的界是紧的。     

Exact and Parameterized Algorithms for Max Internal Spanning Tree

We consider the $\mathcal{NP}$ -hard problem of finding a spanning tree with a maximum number of internal vertices. This problem is a generalization of the famous Hamiltonian Path problem. Our dynamic-programming algorithms for general and degree-bounded graphs have running times of the form $\mathcal{O}^{*}(c^{n})$ with c≤2. For graphs with bounded degree, c<2. The main result, however, is a branching algorithm for graphs with maximum degree three. It only needs polynomial space and has a running time of $\mathcal{O}(1.8612^{n})$ when analyzed with respect to the number of vertices. We also show that its running time is $2.1364^{k} n^{\mathcal{O}(1)}$ when the goal is to find a spanning tree with at least k internal vertices. Both running time bounds are obtained via a Measure & Conquer analysis, the latter one being a novel use of this kind of analysis for parameterized algorithms.     

The Parameterized Complexity of the Shared Center Problem

Recently, the shared center (SC) problem has been proposed as a mathematical model for inferring the allele-sharing status of a given set of individuals using a database of confirmed haplotypes as reference. The problem was proved to be NP-complete and a ratio-2 polynomial-time approximation algorithm was designed for its minimization version (called the closest shared center (CSC) problem). In this paper, we consider the parameterized complexity of the SC problem. First, we show that the SC problem is W[1]-hard with parameters d and n, where d and n are the radius and the number of (diseased or normal) individuals in the input, respectively. Then, we present two asymptotically optimal parameterized algorithms for the problem and apply them to linkage analysis.     

Approximating the Degree-Bounded Minimum Diameter Spanning Tree Problem

We consider the problem of finding a minimum diameter spanning tree with maximum node degree $B$ in a complete undirected edge-weighted graph. We provide an $O(\sqrt{\log_Bn})$-approximation algorithm for the problem. Our algorithm is purely combinatorial, and relies on a combination of filtering and divide and conquer.     

Approximating the Degree-Bounded Minimum Diameter Spanning Tree Problem

We consider the problem of finding a minimum diameter spanning tree with maximum node degree $B$ in a complete undirected edge-weighted graph. We provide an $O(\sqrt{\log_Bn})$-approximation algorithm for the problem. Our algorithm is purely combinatorial, and relies on a combination of filtering and divide and conquer.     

On the History of the Minimum Spanning Tree Problem

It is standard practice among authors discussing the minimum spanning tree problem to refer to the work of Kruskal(1956) and Prim (1957) as the sources of the problem and its first efficient solutions, despite the citation by both of Boruvka (1926) as a predecessor. In fact, there are several apparently independent sources and algorithmic solutions of the problem. They have appeared in Czechoslovakia, France, and Poland, going back to the beginning of this century. We shall explore and compare these works and their motivations, and relate them to the most recent advances on the minimum spanning tree problem.     

一种求解最小生成树问题的算法

基于节点合并和反向追踪的思想,提出一种求解最小生成树问题的算法。该算法依据网络邻接矩阵,将与源节点相邻的节点逐步合并为新的源节点,使网络中的所有节点合并为一个点,借助引入的前点标号数组得到网络的最小生成树,对算法正确性与算法复杂度进行分析。将该算法应用于某高速公路网工程建设方案,结果证明了算法的有效性。     

Improved Heuristics for the Minimum Label Spanning Tree Problem

Given a connected, undirected graph G whose edges are labeled, the minimum label (or labeling) spanning tree (MLST) problem seeks a spanning tree on G with the minimum number of distinct labels. Maximum vertex covering algorithm (MVCA) is a well-known heuristic for the MLST problem. It is very fast and performs reasonably well. Recently, we developed a genetic algorithm (GA) for the MLST problem. The GA and MVCA are similarly fast but the GA outperforms the MVCA. In this paper, we present four modified versions of MVCA, as well as a modified GA. These modified procedures generate better results, but are more expensive computationally. The modified GA is the best performer with respect to both accuracy and running time     

Parameterized Random Complexity

The classes W[P] and W[1] are parameterized analogues of NP in that they can be characterized by machines with restricted existential nondeterminism. These machine characterizations give rise to two natural notions of parameterized randomized algorithms that we call W[P]-randomization and W[1]-randomization. This paper develops the corresponding theory.     

Parameterized Proof Complexity

We propose a proof-theoretic approach for gaining evidence that certain parameterized problems are not fixed-parameter tractable. We consider proofs that witness that a given propositional formula cannot be satisfied by a truth assignment that sets at most k variables to true, considering k as the parameter (we call such a formula a parameterized contradiction). One could separate the parameterized complexity classes FPT and W[SAT] by showing that there is no fpt-bounded parameterized proof system for parameterized contradictions, i.e., that there is no proof system that admits proofs of size f(k)n ^O(1) where f is a computable function and n represents the size of the propositional formula. By way of a first step, we introduce the system of parameterized tree-like resolution and show that this system is not fpt-bounded. Indeed, we give a general result on the size of shortest tree-like resolution proofs of parameterized contradictions that uniformly encode first-order principles over a universe of size n. We establish a dichotomy theorem that splits the exponential case of Riis’s complexity gap theorem into two subcases, one that admits proofs of size f(k)n ^O(1) and one that does not. We also discuss how the set of parameterized contradictions may be embedded into the set of (ordinary) contradictions by the addition of new axioms. When embedded into general (DAG-like) resolution, we demonstrate that the pigeonhole principle has a proof of size 2 ^k n ². This contrasts with the case of tree-like resolution where the embedded pigeonhole principle falls into the “non-FPT” category of our dichotomy.     

一种新的求解最小生成树问题的DNA算法

基于生化反应的生物智能计算是现阶段计算领域研究的热点,DNA计算是通过DNA分子之间的生化反应来进行计算的一种计算模式,凭借运算巨大的并行性和海量存储的优势,DNA计算在解决复杂运算问题方面的计算能力显而易见。设计了一种利用DNA计算来求解图的最小生成树的计算模型,采用一种特殊的编码方式来对顶点,边和权值进行编码,并且描述了MSTP解的计算过程。     

求解度约束最小生成树的新的遗传算法

针对度约束最小生成树问题的特征,设计了一种新的编码方式,并在此基础上提出了一个新遗传算法来求解该问题。该算法采用新的启发式杂交算子、变异算子和局部搜索算子,以概率1收敛到全局最优解。数值实验表明该算法优于文中提出的其他4种算法。     

一种求解度约束最小生成树问题的优化算法

为求解大规模结点度约束最小生成树问题,提出一种带有嫁接和剪接算子操作的优化算法.通过借鉴花草果树种植技术,建立一种以基本遗传算子为基础、带有加速和调节算子作为激励的进化计算体系;嫁接以一种贪婪的思想加速搜索,按收益最大化原则进行剪接.对可能陷入局部极值引起冲突的现象及冲突检测的方法进行分析,并提出了冲突的若干解决方法.针对DCMST问题求解中的复杂性,提出了几种有效的嫁接和剪接的策略,并对算法的收敛性和计算复杂度进行了分析.通过该算法对结点数为50~500之间的Euclidean问题和按均匀随机方式产生的non-Euclidean度约束最小生成树问题进行求解.与现有文献的实验结果对比表明,该方法在求解最好解的精度和收敛速度上均有一定的优势.     

求解多目标最小生成树问题的改进算法

多目标最小生成树问题是典型的NP问题,Zhou和Gen提出了一种用于计数多目标最小生成树问题的所有非劣最优最小生成树的算法,但该算法无法保证能够找到所有非劣最优最小生成树.针对此问题,提出一种改进的计数算法,并定性说明改进算法能够找到问题的所有非劣最优最小生成树.改进算法在进行子树剔除时增加了一些条件.模拟实验结果表明,改进后的计数算法能够找到所有的非劣最优解.这也说明该算法具有应用的潜力.     

最小生成树算法在旅行商问题中的应用

如何在n个顶点之间的1/2(n-1)!巡回路径中选择距离最短的,这是一个典型的组合优化问题,也是解决旅行商问题的根本。在最小生成树的基本思想上进行了改进,成功地解决了旅行商问题。     

Parameterized Complexity and Local Search Approaches for the Stable Marriage Problem with Ties

We consider the variant of the classical Stable Marriage problem where preference lists can be incomplete and may contain ties. In such a setting, finding a stable matching of maximum size is NP-hard. We study the parameterized complexity of this problem, where the parameter can be the number of ties, the maximum or the overall length of ties. We also investigate the applicability of a local search algorithm for the problem. Finally, we examine the possibilities for giving an FPT algorithm or an FPT approximation algorithm for finding an egalitarian or a minimum regret matching.     

模糊权值网络最小生成树问题的矩阵算法

通过研究模糊权值网络中的最小生成树问题,使用基于模糊数的结构元加权序和经典最小生成树问题的改进权矩阵法,本文提出一种求解边权值为三角模糊数的模糊权值网络最小生成树问题的矩阵算法,并对算法的复杂度和正确性进行分析。通过实例验证了该算法的有效性。     

一种新的求解度约束最小生成树的遗传算法

染色体编码是遗传算法的关键内容,编码的优劣并直接影响算法的性能.提出了基于过程控制的生成树编码方法--PC编码.PC码为定长的整数向量,使用PC编码求解特定生成树问题时,首先选定的一个有效算法,并将修改为可控算法,然后用编码向量控制算法的运行过程,从面得到唯一生成树.为了求解度约束最小生成树(DCMST)问题,在D-Prim算法的基础上,设计r过程可控的度约束生成树构造PC-Prim算法.给出了以PC-Prim算法作为译码器的求解DC-MST问题的遗传算法.仿真结果表明遗传算法求解精度和运行时间均优于参与其他算法.